Conditional Association, Essential Independence and Monotone Unidimensional Item Response Models

Abstract

We consider two recent approaches to characterizing the manifest probabilities of a strictly unidimensional latent variable representation (one satisfying local independence and response curve monotonicity with respect to a unidimensional latent variable) for binary response variables, such as those arising from the dichotomous scoring of items on standardized achievement and aptitude tests. Holland and Rosenbaum showed that conditional association is a necessary condition for strict unidimensionality; and Stout treated the class of essentially unidimensional models, in which the latent variable may be consistently estimated as the length of the response sequence grows using the proportion of positive responses. Of particular concern are strictly unidimensional representations that are minimally useful in the sense that: (1) the latent variable can be consistently estimated from the responses; (2) the regression of proportion of positive responses on the latent variable is monotone; and (3) the latent variable is not constant in the population. We introduce two new conditions, a negative association condition and a natural monotonicity condition on the empirical response curves, that help link strict unidimensionality with the conditional association and essential unidimensionality approaches. These conditions are illustrated with a partial characterization of useful, strictly unidimensional representations.